# Some simpler statistical tests for rejecting outliers in quantitative data*

### Some simpler statistical tests for rejecting outliers in quantitative data*

Large data sets (N>100):

For small data sets:

1

Rule of Huge Error: If you have a single outlier, then you can discard it with 98% confidence if any of the following conditions are met.

| |

5 8 6

8 14 5

15 4

Dixon’s Q-test: If you have a single outlier, and your data has a normal distribution, then you can discard the outlier if . Order the data values in increasing or decreasing order, such that the outlier is the final data point (x

N

).

3 7

8 10

11 13

*Compiled from:

-Personal webpage of Prof. James K. Hardy, Dept. of Chemistry, University of Akron, “Statistical Treatment of Data” at http://ull.chemistry.uakron.edu/analytical/Statistics/. This has good notes for basic statistics and refers to specific tests for the rejection of data and discusses large and small sample sets.

-“Dixon's Q-test: Detection of a single outlier”, which includes an Applet for doing Q-test calculations and a brief discussion on rejecting data from small data sets, on the University of Athen’s Department of Chemistry website at http://www.chem.uoa.gr/Applets/AppletQtest/Appl_Qtest2.html.

Note: although much of the department’s website is in Greek, this page is in English.

-“Statistical Treatment of Analytical Data: Outliers (Chapter 6)” by Z.B. Alfassi, Z. Boger and Y. Ronen. CRC Press: 2005. This chapter is available for reading through Google books if your library doesn’t have a copy.

Qcrit Values for Dixon's Q-test Outliers

Data points

N

5

6

3

4

7

8

9

10

11

12

13

0.5

0.994

0.926

0.821

0.740

0.680

0.725

0.677

0.639

0.713

0.675

0.649

Risk of false rejection (%)

1 5

0.988

0.889

0.780

0.698

0.637

0.683

0.635

0.597

0.679

0.642

0.615

0.941

0.765

0.642

0.560

0.507

0.554

0.512

0.477

0.576

0.546

0.521

10

0.886

0.679

0.557

0.482

0.434

0.479

0.441

0.409

0.517

0.490

0.467

Grubbs’ T-test: This test can be used to evaluate multiple possible outliers. Start with the furthest outlier, | | , and discard it if T > Tcrit.

| |

Tcrit Values for Grubbs' T-test for Outliers

Data points Risk of false rejection (%)

N

3

0.1

1.155

0.5

1.155

1

1.155

5

1.153

6

7

8

4

5

1.496

1.780

2.011

2.201

2.358

1.496

1.764

1.973

2.139

2.274

1.492

1.749

1.944

2.097

2.221

1.463

1.672

1.822

1.938

2.032

9

10

15

20

25

50

100

2.492

2.606

2.997

3.230

3.389

3.789

4.084

2.387

2.482

2.806

3.001

3.135

3.483

3.754

2.323

2.410

2.705

2.884

3.009

3.336

3.600

2.110

2.176

2.409

2.557

2.663

2.956

3.207

10

1.148

1.425

1.602

1.729

1.828

1.909

1.977

2.036

2.247

2.385

2.486

2.768

3.017

If you discard the outlier, and suspect others, then recalculate and s in order to evaluate the next furthest point.